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Week 1 - Graded Assignment 1

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22 min read
Course: Jan 2026 - Mathematics I
Topic: Set Theory and Relations | Marks: 1

Question 1

A survey was conducted on pollution of 525 ponds across some cities. It was found that 230 ponds are polluted by fertilisers (F)(F), 245 ponds are polluted by pesticides (P)(P) and 257 ponds are polluted by pharmaceutical products (Ph)(Ph). 100 ponds are polluted by fertilisers and pesticides, 82 ponds are polluted by fertilisers and pharmaceutical products, 77 ponds are polluted by pesticides and pharmaceutical products.
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Define a relation on the set of 525 ponds such that two ponds are related if both are polluted by fertilisers and pharmaceutical products. Which of the following is/are true?
Type: MSQ | Marks: 1 | Topic: Relations
MCQ
Set Theory and Relations
3 Unit Assessment
A
Relation is reflexive.
B
Relation is transitive.
C
Relation is symmetric.
D
This is an equivalence relation.
Topic: Identity Relations and Functions | Marks: 1

Question 2

Consider the relation R={(x,y)∣x−y=0}⊂R×RR= \{(x, y) \mid x-y = 0 \} \subset \mathbb{R} \times \mathbb{R} on the set R\mathbb{R}. Which of the following is/are true?
Type: MSQ | Marks: 1 | Topic: Relations
MCQ
Set Theory and Relations
3 Unit Assessment
A
RR is a transitive relation.
B
RR is a function.
C
RR is not an equivalence relation.
D
RR is a reflexive relation.
E
RR is a symmetric relation.
Topic: Injective (One-One) Functions | Marks: 1

Question 3

Which of the following relations is/are one-one function?
Type: MCQ | Marks: 1 | Topic: Functions
MCQ
Set Theory and Relations
3 Unit Assessment
A
R1={(x,y)∣x,y∈R,x+y>2}R_1 = \{(x, y) \mid x, y \in \mathbb{R}, x+y>2 \}
B
R2={(x,y)∣x,y∈R,x>y}R_2 = \{(x, y) \mid x, y \in \mathbb{R}, x>y \}
C
R3={(x,y)∣x,y∈R, x+y=12}R_3 = \{(x,y) \mid x,y \in \mathbb{R},\ x + y = 12\}
D
R4={(x,y)∣x,y∈R,y=x2}R_4 = \{(x, y) \mid x, y \in \mathbb{R}, y=x^2 \}
Topic: Surjective (Onto) and Piecewise Functions | Marks: 1

Question 4

Which of the following is/are true?
Type: MSQ | Marks: 1 | Topic: Functions
MCQ
Set Theory and Relations
3 Unit Assessment
A
Function f:N→Rf: \mathbb{N} \to \mathbb{R} such that f(x)=x2f(x) = x^2 is not onto.
B
Relation R={(1,1),(1,2),(3,1)}R = \{(1,1),(1,2), (3,1) \} on a set A={1,2,3}A = \{1, 2, 3\} is a function.
C
Function f:R→Rf: \mathbb{R} \to \mathbb{R} such that f(x)=∣x∣f(x) = |x| is not one-one.
D
Function f:R→Rf: \mathbb{R} \to \mathbb{R} such that f(x)=(x−1)2f(x) = (x-1)^2 is one-one and onto.
Topic: Set Cardinality and Regions | Marks: 1

Question 5

Consider the following sets,
A={x∣x∈N}A = \{x \mid x \in\mathbb{N}\}
B={x∣−5<x<105,x∈R}B = \{x \mid -5< x< 105, x \in \mathbb{R} \}
C={x∣x is a rational number,10<x≤80}C = \{x \mid x~ \text{is a rational number}, 10< x \leq 80 \}
Find the cardinality of the set (A−C)∩B(A - C) \cap B.
Type: NAT | Marks: 1 | Topic: Set Theory
MCQ
3 Unit Assessment
Topic: Principle of Inclusion-Exclusion (PIE) | Marks: 1

Question 6

A survey was conducted on pollution of 525 ponds across some cities. It was found that 230 ponds are polluted by fertilisers (F)(F), 245 ponds are polluted by pesticides (P)(P) and 257 ponds are polluted by pharmaceutical products (Ph)(Ph). 100 ponds are polluted by fertilisers and pesticides, 82 ponds are polluted by fertilisers and pharmaceutical products, 77 ponds are polluted by pesticides and pharmaceutical products.
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Find the number of ponds polluted by all types of water contaminants.
Type: NAT | Marks: 1 | Topic: Set Theory
MCQ
3 Unit Assessment
Topic: Modeling Family Trees as Functions | Marks: 1

Question 7

Mahesh has four sons (Shubh, Rabi, Mahendra, and Rajat). Shubh has two sons (Yashubh and Navrtna). Rabi has two sons named Rathi and Rakesh.
Let RR be "are cousins" and SS be "is son of". f=S∩(P×Q)f = S \cap (P \times Q).
Which of the following options are correct?
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Type: MSQ | Marks: 1 | Topic: Relations & Functions
MCQ
Set Theory and Relations
3 Unit Assessment
A
f:P→Qf: P \to Q is a function, where P={Yashubh, Navrtna, Rathi, Rakesh, Mahesh}P = \{\text{Yashubh, Navrtna, Rathi, Rakesh, Mahesh}\} and Q={ Shubh, Rabi, Mahendra, Rajat}Q=\{\text{ Shubh, Rabi, Mahendra, Rajat}\}.
B
If f:P→Qf: P \to Q is a function, where P={ Yashubh, Navrtna, Rathi, Rakesh}P=\{\text{ Yashubh, Navrtna, Rathi, Rakesh}\} and Q={Shubh, Rabi, Mahendra, Rajat}Q=\{\text{Shubh, Rabi, Mahendra, Rajat}\}, then ff is one-one.
C
If f:P→Qf: P \to Q is a function, where P={Yashubh, Navrtna, Rathi, Rakesh}P=\{\text{Yashubh, Navrtna, Rathi, Rakesh}\} and Q={ Shubh, Rabi}Q=\{ \text{ Shubh, Rabi}\}, then ff is onto.
D
If f:P→Qf: P \to Q is a function, where P={Yashubh, Rathi}P=\{\text{Yashubh, Rathi}\} and Q={Shubh, Rabi}Q=\{\text{Shubh, Rabi}\}, then ff is bijective.
Topic: Domains and Codomains | Marks: 1

Question 8

Consider the table of materials and dielectric constants.
MaterialDielectric constant
Air1
Vaccum2
Paper3
Glass8
Nerve membrane7
Silicon13
Let ff be a function from the set of materials to the set of dielectric constants. Which of the following is true?
Type: MCQ | Marks: 1 | Topic: Functions
MCQ
Set Theory and Relations
3 Unit Assessment
A
ff is neither one to one nor onto.
B
ff is one to one but not onto.
C
ff is onto but not one to one.
D
ff is bijective.
Topic: Symmetric Difference of Sets | Marks: 1

Question 9

Consider the following sets.
  • A={x∈N∣x mod 2=0 and 1≤x≤10}A = \lbrace x\in \mathbb{N}\mid x \text{ mod } 2 = 0 \text { and } 1\leq x \leq 10 \rbrace
  • B={x∈N∣x mod 5=0 and 6≤x≤25}B = \lbrace x\in \mathbb{N}\mid x \text{ mod } 5 = 0 \text { and } 6\leq x \leq 25 \rbrace
  • C={x∈N∣x mod 7=0 and 7≤x≤29}C = \lbrace x\in \mathbb{N}\mid x \text{ mod } 7 = 0 \text { and } 7\leq x \leq 29 \rbrace
What is the cardinality of (A∖(B∪C))∪(B∖(C∪A))∪(C∖(B∪A))\big (A\setminus (B\cup C) \big) \cup \big (B\setminus (C\cup A)\big )\cup \big (C\setminus (B\cup A)\big )
Type: NAT | Marks: 1 | Topic: Set Theory
MCQ
3 Unit Assessment
Topic: Multi-set Overlap Word Problems | Marks: 1

Question 10

In 180 people survey: 95 watched Dabangg, 100 watched Avatar, 100 watched RRR. Intersections: Dabangg ∩\cap Avatar = 50, Avatar ∩\cap RRR = 40, Dabangg ∩\cap RRR = 55. If all 180 people watched at least one movie, find the number of people who watched only RRR and Avatar but not Dabangg.
Type: NAT | Marks: 1 | Topic: Set Theory
MCQ
3 Unit Assessment

Question 11

Family Tree relations RR ("are cousins") and SS ("is son of"). Mahesh has four sons (Shubh, Rabi, Mahendra, and Rajat). Shubh has two sons (Yashubh and Navrtna). Rabi has two sons named Rathi and Rakesh.
Which of the following options are correct?
Type: MSQ | Marks: 1 | Topic: Relations
MCQ
Set Theory and Relations
3 Unit Assessment
A
RR is an equivalence relation.
B
(Rathi, Navrtna) ∈R\in R
C
SS is an equivalence relation.
D
(Mahesh, Rajat) ∈S\in S.
E
(Yashubh, Rathi) ∈R\in R and (Rathi, Yashubh) ∈R\in R.
F
SS is symmetric relation.

Question 12

Family Tree: Let mm be the number of elements in RR (cousins) and nn be the number of elements in SS (son of). Find m+nm+n. (Consider the set consisting of all mentioned people).
Type: NAT | Marks: 1 | Topic: Relations
MCQ
3 Unit Assessment

Question 13

Let A=QA = \mathbb{Q} and B=ZB = \mathbb{Z}. Consider the function f:A→Bf: A \to B defined as f(p/q)=p−qf(p/q) = p - q where gcd(p,q)=1gcd(p, q) = 1 and q>0q > 0. Which of the following is true?
Type: MCQ | Marks: 1 | Topic: Functions
MCQ
Set Theory and Relations
3 Unit Assessment
A
ff is one to one but not onto.
B
ff is neither one to one nor onto.
C
ff is onto but not one to one.
D
ff is bijective.

Question 14

Match the following sets: i. Finitei. \text{ Finite} ii. Infiniteii. \text{ Infinite} iii. Singletoniii. \text{ Singleton} iv. Emptyiv. \text{ Empty}
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(A) {x∣x2=25,x∈N}\{x \mid x^2 = 25, x \in \mathbb{N} \} (B) {x∣x2+1=0,x∈R}\{x \mid x^2 + 1 = 0, x \in \mathbb{R} \} (C) {x∣x2−x−6=0,x∈Z}\{x \mid x^2 - x - 6 = 0, x \in \mathbb{Z} \} (D) {x∣x+2>0,x∈R}\{x \mid x+2 > 0, x \in \mathbb{R} \}
Type: MCQ | Marks: 1 | Topic: Set Theory
MCQ
Set Theory and Relations
3 Unit Assessment
A
A→iii,B→iv,C→i,D→iiA \to iii, B \to iv, C \to i, D \to ii
B
A→iii,B→i,C→iv,D→iiA \to iii, B \to i, C \to iv, D \to ii
C
A→iii,B→iv,C→i,D→iiA \to iii, B \to iv, C \to i, D \to ii (Wait, same as A)
D
A→iii,B→ii,C→iv,D→iA \to iii, B \to ii, C \to iv, D \to i

Question 15

If f1f_1 and f2f_2 are two functions with domains D1D_1 and D2D_2 respectively, then the domain of f1+f2f_1 + f_2 is:
Type: MCQ | Marks: 1 | Topic: Functions
MCQ
Set Theory and Relations
3 Unit Assessment
A
D1∪D2D_1 \cup D_2
B
D1∖D2D_1 \setminus D_2
C
D2∖D1D_2 \setminus D_1
D
D1∩D2D_1 \cap D_2

Question 16

Let f,g:R→Rf, g: \mathbb{R} \to \mathbb{R} be defined as f(x)={xx≤0 0x>0f(x) = \begin{cases} x & x \leq 0 \ 0 & x > 0 \end{cases} g(x)={0x≤0 xx>0g(x) = \begin{cases} 0 & x \leq 0 \ x & x > 0 \end{cases} Then f+gf + g is:
Type: MSQ | Marks: 1 | Topic: Functions
MCQ
Set Theory and Relations
3 Unit Assessment
A
ff is one-to-one but not onto.
B
ff is neither.
C
gg is onto but not one-to-one.
D
f+gf+g is bijective.

Question 17

Consider the relations R1={(a,b)∈N×N∣b=a+1}R_1 = \{(a, b) \in \mathbb{N} \times \mathbb{N} \mid b = a + 1 \} R2={(a,b)∈N×N∣b≥a}R_2 = \{(a, b) \in \mathbb{N} \times \mathbb{N} \mid b \geq a \} Which of the following is true?
Type: MSQ | Marks: 1 | Topic: Relations
MCQ
Set Theory and Relations
3 Unit Assessment
A
R1R_1 is reflexive.
B
R2R_2 is symmetric.
C
R1R_1 is neither symmetric nor transitive.
D
R2R_2 is both reflexive and transitive.

Question 18

Let A={3,6,9,...}A = \{3, 6, 9, ... \} (set of positive multiples of 3). Let f:N→Af: \mathbb{N} \to A be defined as f(n)=6nf(n) = 6n. Is ff bijective?
Type: MSQ | Marks: 1 | Topic: Functions
MCQ
Set Theory and Relations
3 Unit Assessment
A
ff is one-to-one but not onto.
B
ff is neither.
C
ff is onto but not one-to-one.
D
ff is bijective.

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